let i be Element of NAT ; for G being Go-board st 1 <= i & i + 1 <= len G holds
((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]| in Int (cell G,i,(width G))
let G be Go-board; ( 1 <= i & i + 1 <= len G implies ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]| in Int (cell G,i,(width G)) )
assume that
A1:
1 <= i
and
A2:
i + 1 <= len G
; ((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]| in Int (cell G,i,(width G))
set r1 = (G * i,(width G)) `1 ;
set s1 = (G * i,(width G)) `2 ;
set r2 = (G * (i + 1),(width G)) `1 ;
width G <> 0
by GOBOARD1:def 5;
then A3:
1 <= width G
by NAT_1:14;
i < len G
by A2, NAT_1:13;
then A4:
Int (cell G,i,(width G)) = { |[r,s]| where r, s is Real : ( (G * i,1) `1 < r & r < (G * (i + 1),1) `1 & (G * 1,(width G)) `2 < s ) }
by A1, Th28;
width G <> 0
by GOBOARD1:def 5;
then A5:
1 <= width G
by NAT_1:14;
i < i + 1
by XREAL_1:31;
then A6:
(G * i,(width G)) `1 < (G * (i + 1),(width G)) `1
by A1, A2, A5, GOBOARD5:4;
then
((G * i,(width G)) `1 ) + ((G * i,(width G)) `1 ) < ((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 )
by XREAL_1:8;
then A7:
(1 / 2) * (((G * i,(width G)) `1 ) + ((G * i,(width G)) `1 )) < (1 / 2) * (((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 ))
by XREAL_1:70;
A8:
i < len G
by A2, NAT_1:13;
then A9:
(G * 1,(width G)) `2 = (G * i,(width G)) `2
by A1, A3, GOBOARD5:2;
then A10:
(G * 1,(width G)) `2 < ((G * i,(width G)) `2 ) + 1
by XREAL_1:31;
A11:
1 <= i + 1
by NAT_1:11;
then
(G * 1,(width G)) `2 = (G * (i + 1),(width G)) `2
by A2, A3, GOBOARD5:2;
then
( G * i,(width G) = |[((G * i,(width G)) `1 ),((G * i,(width G)) `2 )]| & G * (i + 1),(width G) = |[((G * (i + 1),(width G)) `1 ),((G * i,(width G)) `2 )]| )
by A9, EUCLID:57;
then
( (1 / 2) * (((G * i,(width G)) `2 ) + ((G * i,(width G)) `2 )) = (G * i,(width G)) `2 & (G * i,(width G)) + (G * (i + 1),(width G)) = |[(((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 )),(((G * i,(width G)) `2 ) + ((G * i,(width G)) `2 ))]| )
by EUCLID:60;
then
(1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G))) = |[((1 / 2) * (((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 ))),((G * i,(width G)) `2 )]|
by EUCLID:62;
then A12:
((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]| = |[(((1 / 2) * (((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 ))) + 0 ),(((G * i,(width G)) `2 ) + 1)]|
by EUCLID:60;
((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 ) < ((G * (i + 1),(width G)) `1 ) + ((G * (i + 1),(width G)) `1 )
by A6, XREAL_1:8;
then
(1 / 2) * (((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 )) < (1 / 2) * (((G * (i + 1),(width G)) `1 ) + ((G * (i + 1),(width G)) `1 ))
by XREAL_1:70;
then A13:
(1 / 2) * (((G * i,(width G)) `1 ) + ((G * (i + 1),(width G)) `1 )) < (G * (i + 1),1) `1
by A2, A11, A3, GOBOARD5:3;
(G * i,1) `1 = (G * i,(width G)) `1
by A1, A8, A3, GOBOARD5:3;
hence
((1 / 2) * ((G * i,(width G)) + (G * (i + 1),(width G)))) + |[0 ,1]| in Int (cell G,i,(width G))
by A12, A7, A13, A10, A4; verum